β

Hamiltonian (quantum mechanics)

In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases. It is usually denoted by H, also Ȟ or Ĥ. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

Contents

The Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system. For different situations or number of particles, the Hamiltonian is different since it includes the sum of kinetic energies of the particles, and the potential energy function corresponding to the situation.

which allows one to apply the Hamiltonian to systems described by a wave functionΨ(r, t). This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.

One can also make substitutions to certain variables to fit specific cases, such as some involving electromagnetic fields.

However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:

where M denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many electron atoms (see below).

For N interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function V is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.

For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle,[1] that is

where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation - in practice the particles are almost always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they interact with each other by Coulomb interaction (electrostatic force), as shown below.

This equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation, which is one of the reasons H is also called the Hamiltonian. Given the state at some initial time (t = 0), we can solve it to obtain the state at any subsequent time. In particular, if H is independent of time, then

The exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in H. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a functional calculus is required. In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.

By the *-homomorphism property of the functional calculus, the operator

The eigenkets (eigenvectors) of H, denoted |a⟩{\displaystyle \left|a\right\rangle }, provide an orthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted {Ea}, solving the equation:

From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.[clarification needed]

Following are expressions for the Hamiltonian in a number of situations.[2] Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function - importantly space and time dependence. Masses are denoted by m, and charges by q.

where Ixx, Iyy, and Izz are the moment of inertia components (technically the diagonal elements of the moment of inertia tensor), and J^x{\displaystyle {\hat {J}}_{x}\,\!}, J^y{\displaystyle {\hat {J}}_{y}\,\!} and J^z{\displaystyle {\hat {J}}_{z}\,\!} are the total angular momentum operators (components), about the x, y, and z axes respectively.

For a charged particle q in an electromagnetic field, described by the scalar potentialφ and vector potentialA, there are two parts to the Hamiltonian to substitute for.[1] The momentum operator must be replaced by the kinetic momentum operator, which includes a contribution from the A field:

In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operatorUcommutes with the Hamiltonian. To see this, suppose that |a⟩{\displaystyle |a\rangle } is an energy eigenket. Then U|a⟩{\displaystyle U|a\rangle } is an energy eigenket with the same eigenvalue, since

Since U is nontrivial, at least one pair of |a⟩{\displaystyle |a\rangle } and U|a⟩{\displaystyle U|a\rangle } must represent distinct states. Therefore, H has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.

The existence of a symmetry operator implies the existence of a conserved observable. Let G be the Hermitian generator of U:

U=I−iϵG+O(ϵ2){\displaystyle U=I-i\epsilon G+O(\epsilon ^{2})\,}

It is straightforward to show that if U commutes with H, then so does G:

Hamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states {|n⟩}{\displaystyle \left\{\left|n\right\rangle \right\}}, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

⟨n′|n⟩=δnn′{\displaystyle \langle n'|n\rangle =\delta _{nn'}}

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at time t, |ψ(t)⟩{\displaystyle \left|\psi \left(t\right)\right\rangle }, can be expanded in terms of these basis states:

The coefficients an(t) are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.

The expectation value of the Hamiltonian of this state, which is also the mean energy, is

where the last step was obtained by expanding |ψ(t)⟩{\displaystyle \left|\psi \left(t\right)\right\rangle } in terms of the basis states.

Each an(t) actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use an(t) and its complex conjugatean*(t). With this choice of independent variables, we can calculate the partial derivative

which is precisely the form of Hamilton's equations, with the an{\displaystyle a_{n}}s as the generalized coordinates, the πn{\displaystyle \pi _{n}}s as the conjugate momenta, and ⟨H⟩{\displaystyle \langle H\rangle } taking the place of the classical Hamiltonian.